Class 10 Maths — Important Questions with Answers

15 concept-first Class 10 Maths questions across 13 chapters — each with a clear model answer, the why behind it, and a memory trick to make it stick. NCERT-aligned and free. Build the base now, and your board exams, NEET and JEE feel a whole lot easier later.

15 questions•13 chapters•Concept-first•NCERT-aligned•For boards, NEET & JEE

Real Numbers (1)Polynomials (1)Pair of Linear Equations in Two Variables (1)Quadratic Equations (1)Arithmetic Progressions (2)Triangles (1)Coordinate Geometry (1)Introduction to Trigonometry (2)Some Applications of Trigonometry (1)Circles (1)Surface Areas and Volumes (1)Statistics (1)Probability (1)

Real Numbers1 question

MathematicsEuclid's division algorithmmedium

Find the HCF of 96 and 404 using Euclid's division algorithm.

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What it is

Euclid's algorithm finds the HCF by repeated division until the remainder becomes zero.

Answer

Apply repeated division: 404 = 96 x 4 + 20; 96 = 20 x 4 + 16; 20 = 16 x 1 + 4; 16 = 4 x 4 + 0. The last non-zero remainder is 4, so HCF(96, 404) = 4.

a = bq + r, then HCF(a,b) = HCF(b,r)

•404 = 96 x 4 + 20
•96 = 20 x 4 + 16
•20 = 16 x 1 + 4
•16 = 4 x 4 + 0 -> HCF = 4

Why learn this

HCF simplifies fractions, shares things equally and even underpins computer cryptography (RSA).

💡 Memory trick

Divide, take the remainder, divide again. Last non-zero remainder = HCF.

Polynomials1 question

MathematicsZeroes of a quadraticeasy

Find the zeroes of the polynomial x^2 - 7x + 10 and verify the relationship between the zeroes and the coefficients.

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What it is

The zeroes of a quadratic are where it equals zero; their sum is -b/a and product is c/a.

Answer

Factorise: x^2 - 7x + 10 = (x - 2)(x - 5), so the zeroes are 2 and 5. Sum of zeroes = 2 + 5 = 7 = -(-7)/1 = -b/a. Product of zeroes = 2 x 5 = 10 = 10/1 = c/a. Both relations hold.

Sum = -b/a ; Product = c/a

•(x - 2)(x - 5) -> zeroes 2, 5
•Sum = 7 = -b/a
•Product = 10 = c/a

Why learn this

Finding roots is how we solve projectile paths, profit models and design problems in engineering.

💡 Memory trick

Sum = -b/a (has the minus), Product = c/a (plain).

Pair of Linear Equations in Two Variables1 question

MathematicsSubstitution methodmedium

Solve the pair of equations: 2x + 3y = 13 and x - y = -1.

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What it is

Two linear equations meet at one point - the (x, y) that satisfies both at once.

Answer

From the second equation, x = y - 1. Substitute into the first: 2(y - 1) + 3y = 13, so 5y - 2 = 13, giving 5y = 15 and y = 3. Then x = y - 1 = 2. Solution: x = 2, y = 3.

Substitution: express one variable, substitute into the other

•x = y - 1 from second equation
•2(y-1) + 3y = 13 -> 5y = 15
•y = 3, x = 2

Why learn this

It solves 'two unknowns' problems: prices, mixtures, speed-time and business break-even.

💡 Memory trick

Substitution: make one variable the subject, then plug it into the other equation.

Quadratic Equations1 question

MathematicsNature of rootsmedium

Find the roots of x^2 - 5x + 6 = 0 and state the nature of the roots using the discriminant.

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What it is

The discriminant D = b^2 - 4ac reveals the nature of a quadratic's roots before you solve it.

Answer

Factorise: (x - 2)(x - 3) = 0, so the roots are x = 2 and x = 3. The discriminant D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Since D > 0 and is a perfect square, the roots are real, distinct and rational.

D = b^2 - 4ac ; x = (-b +/- sqrt(D)) / 2a

•(x-2)(x-3) = 0 -> roots 2, 3
•D = b^2 - 4ac = 1
•D > 0 -> two distinct real roots

Why learn this

Engineers use it to know at a glance if a design has real solutions (bridges, projectiles).

💡 Memory trick

D > 0 real and distinct; D = 0 real and equal; D < 0 no real roots.

Arithmetic Progressions2 questions

Mathematicsnth termeasy

The first term of an AP is 3 and the common difference is 5. Find its 10th term.

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What it is

In an AP each term rises by a fixed step d; the nth term is a + (n - 1)d.

Answer

The nth term of an AP is a_n = a + (n - 1)d. Here a = 3, d = 5, n = 10, so a_10 = 3 + (10 - 1) x 5 = 3 + 45 = 48.

a_n = a + (n - 1)d

•a_n = a + (n - 1)d
•a = 3, d = 5, n = 10
•a_10 = 48

Why learn this

APs model salaries with fixed raises, seating rows, EMIs and steady savings.

💡 Memory trick

nth term = start + (steps taken) x step = a + (n - 1)d.

MathematicsSum of n termsmedium

Find the sum of the first 20 terms of the AP 2, 5, 8, 11, ...

Reveal answer▾

What it is

The sum of an AP pairs the first and last terms: S = n/2 [2a + (n - 1)d].

Answer

Here a = 2 and d = 3. The sum of n terms is S_n = n/2 [2a + (n - 1)d]. So S_20 = 20/2 [2(2) + 19(3)] = 10 [4 + 57] = 10 x 61 = 610.

S_n = n/2 [2a + (n - 1)d]

•a = 2, d = 3, n = 20
•S_n = n/2 [2a + (n-1)d]
•S_20 = 10 x 61 = 610

Why learn this

It quickly totals things that grow steadily - savings, stacked logs, salaries over years.

💡 Memory trick

Sum = (number of terms / 2) x (2a + (n - 1)d): half the count times the bracket.

Triangles1 question

MathematicsBasic Proportionality Theoremmedium

State the Basic Proportionality Theorem (Thales' theorem).

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What it is

A line parallel to one side of a triangle cuts the other two sides in the same ratio.

Answer

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then it divides those two sides in the same ratio. In triangle ABC, if DE is parallel to BC and meets AB at D and AC at E, then AD/DB = AE/EC.

If DE || BC then AD/DB = AE/EC

•Line parallel to one side of a triangle
•Divides the other two sides in the same ratio
•AD/DB = AE/EC
•Basis of similarity of triangles

Why learn this

It's the basis of similarity, scale drawings, maps and finding heights from shadows.

💡 Memory trick

Parallel line inside a triangle -> equal ratios: AD/DB = AE/EC.

Coordinate Geometry1 question

MathematicsDistance formulaeasy

Find the distance between the points (2, 3) and (5, 7).

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What it is

The distance between two points is the hypotenuse of the right triangle of their coordinate gaps.

Answer

The distance formula is d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Here d = sqrt[(5 - 2)^2 + (7 - 3)^2] = sqrt[9 + 16] = sqrt[25] = 5 units.

d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]

•d = sqrt[(x2-x1)^2 + (y2-y1)^2]
•= sqrt[9 + 16] = sqrt[25]
•Distance = 5 units

Why learn this

GPS, video games, robotics and maps all compute 'how far' with this exact formula.

💡 Memory trick

It's just Pythagoras: sqrt of (x-gap)^2 + (y-gap)^2.

Introduction to Trigonometry2 questions

MathematicsTrigonometric identitymedium

If sin(theta) = 3/5, find cos(theta) and tan(theta), where theta is acute.

Reveal answer▾

What it is

The identity sin^2 + cos^2 = 1 lets you find one trig ratio from another for the same angle.

Answer

Using the identity sin^2(theta) + cos^2(theta) = 1, cos^2(theta) = 1 - (3/5)^2 = 1 - 9/25 = 16/25, so cos(theta) = 4/5 (positive since theta is acute). Then tan(theta) = sin/cos = (3/5) / (4/5) = 3/4.

sin^2(theta) + cos^2(theta) = 1 ; tan = sin/cos

•sin^2 + cos^2 = 1
•cos(theta) = 4/5
•tan(theta) = sin/cos = 3/4

Why learn this

Trig ratios drive navigation, physics waves, construction and computer graphics.

💡 Memory trick

Think of the 3-4-5 triangle: sin 3/5 -> cos 4/5 -> tan 3/4.

MathematicsStandard anglesmedium

Evaluate 2 tan^2(45 degrees) + cos^2(30 degrees) - sin^2(60 degrees).

Reveal answer▾

What it is

The standard angles 0, 30, 45, 60 and 90 degrees have fixed trig values worth memorising.

Answer

Use standard values: tan(45 deg) = 1, cos(30 deg) = sqrt(3)/2, sin(60 deg) = sqrt(3)/2. So the expression = 2(1)^2 + (sqrt(3)/2)^2 - (sqrt(3)/2)^2 = 2 + 3/4 - 3/4 = 2.

tan45 = 1 ; cos30 = sqrt(3)/2 ; sin60 = sqrt(3)/2

•tan45 = 1, cos30 = sin60 = sqrt(3)/2
•2(1) + 3/4 - 3/4
•= 2

Why learn this

They appear in nearly every physics and maths problem - knowing them saves huge time.

💡 Memory trick

For sin, use sqrt(0..4)/2 across 0,30,45,60,90; cos is the same list reversed.

Some Applications of Trigonometry1 question

MathematicsHeights and distancesmedium

The angle of elevation of the top of a tower from a point 30 m from its base is 30 degrees. Find the height of the tower.

Reveal answer▾

What it is

An angle of elevation turns a triangle into a height problem using tan = opposite / adjacent.

Answer

Let the height be h. tan(30 degrees) = h / 30, and tan(30 degrees) = 1/sqrt(3). So h = 30 x (1/sqrt(3)) = 30/sqrt(3) = 10 sqrt(3) m, which is about 17.3 m.

tan(theta) = height / base distance

•tan(angle) = opposite/adjacent = h/30
•tan(30 deg) = 1/sqrt(3)
•h = 10 sqrt(3) m (about 17.3 m)

Why learn this

Surveyors, pilots and builders measure unreachable heights (towers, hills) exactly this way.

💡 Memory trick

Draw the right triangle; tan(angle) = height / base distance.

Circles1 question

MathematicsLength of tangentmedium

A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent drawn from P to the circle.

Reveal answer▾

What it is

A tangent touches a circle at one point and is perpendicular to the radius at that point.

Answer

The tangent is perpendicular to the radius at the point of contact, so the radius, tangent and line OP form a right-angled triangle with OP as the hypotenuse. Length of tangent = sqrt[OP^2 - r^2] = sqrt[13^2 - 5^2] = sqrt[169 - 25] = sqrt[144] = 12 cm.

tangent length = sqrt[OP^2 - r^2]

•Tangent is perpendicular to radius at contact
•Right triangle: OP hypotenuse
•Length = sqrt[13^2 - 5^2] = 12 cm

Why learn this

It's used in gear design, road curves and belt-and-pulley engineering.

💡 Memory trick

Radius, tangent and line to the point form a right triangle: tangent = sqrt(OP^2 - r^2).

Surface Areas and Volumes1 question

MathematicsVolume of a coneeasy

Find the volume of a cone whose base radius is 3 cm and height is 4 cm. (Take pi = 22/7.)

Reveal answer▾

What it is

A cone's volume is exactly one-third of a cylinder with the same base and height.

Answer

Volume of a cone = (1/3) x pi x r^2 x h = (1/3) x (22/7) x 3^2 x 4 = (1/3) x (22/7) x 36 = (22 x 12) / 7 = 264/7, which is about 37.7 cubic cm.

V(cone) = (1/3) pi r^2 h

•V = (1/3) pi r^2 h
•= (1/3)(22/7)(9)(4)
•= 264/7 = about 37.7 cm^3

Why learn this

It's how we measure ice-cream cones, funnels, heaps of grain and tent capacity.

💡 Memory trick

Cone = 1/3 x pi r^2 h - a cone is a third of its cylinder.

Statistics1 question

MathematicsEmpirical relationmedium

The mean of a distribution is 27 and its median is 30. Estimate the mode using the empirical relationship.

Reveal answer▾

What it is

For a moderately skewed data set, the three averages link as Mode = 3 Median - 2 Mean.

Answer

The empirical relationship between the three measures of central tendency is Mode = 3 Median - 2 Mean. Substituting, Mode = 3(30) - 2(27) = 90 - 54 = 36.

Mode = 3 Median - 2 Mean

•Mode = 3 Median - 2 Mean
•= 3(30) - 2(27)
•= 90 - 54 = 36

Why learn this

It estimates the most common value from the other two - handy in surveys and economics.

💡 Memory trick

Mode = 3 Median - 2 Mean ('3M minus 2-Mean gives the Mode').

Probability1 question

MathematicsSimple probabilityeasy

One card is drawn at random from a well-shuffled deck of 52 playing cards. What is the probability that it is a king?

Reveal answer▾

What it is

Probability = favourable outcomes / total outcomes, always a number between 0 and 1.

Answer

There are 4 kings in a deck of 52 cards. Probability = number of favourable outcomes / total outcomes = 4/52 = 1/13.

P(event) = favourable outcomes / total outcomes

•4 kings in 52 cards
•P = favourable/total = 4/52
•= 1/13

Why learn this

It powers weather forecasts, insurance, games, medical trials and AI.

💡 Memory trick

Favourable over Total: 4 kings out of 52 -> 4/52 = 1/13.

Start early. Stay ahead.

A few Class 10 Maths questions a day and the concepts become second nature. Explore every subject for Class 10 in the interactive Foundation bank, or ask the AI tutor any doubt the moment it shows up.

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